L9a45

(Knotscape image)

See the full Thistlethwaite Link Table (up to 11 crossings).

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_18,12,9,11 X_16,14,17,13 X_8,16,5,15 X_14,8,15,7 X_12,18,13,17 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -8, 2, -9}, {8, -1, 6, -5}, {9, -2, 3, -7, 4, -6, 5, -4, 7, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$3 v u -2 v w u + 2 w u -2 u -2 v + 2 v w -3 w + 2$ (db)
Jones polynomial	$- q 5 + 3 q 4 -4 q 3 + 5 q 2 -6 q + 7-4 q -1 + 4 q -2 - q -3 + q -4$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a 4 z -2 + a 4 -2 z 2 a 2 -2 a 2 z -2 -3 a 2 + z 4 + z 2 + z -2 + 2 + z 4 a -2 + z 2 a -2 - z 2 a -4$ (db)
Kauffman polynomial	$z 8 a -2 + z 8 + a z 7 + 4 z 7 a -1 + 3 z 7 a -3 + a 2 z 6 + z 6 a -2 + 3 z 6 a -4 - z 6 + a 3 z 5 + a z 5 -9 z 5 a -1 -8 z 5 a -3 + z 5 a -5 + a 4 z 4 + 2 a 2 z 4 -5 z 4 a -2 -8 z 4 a -4 + 4 z 4 + 7 z 3 a -1 + 5 z 3 a -3 -2 z 3 a -5 -3 a 4 z 2 -6 a 2 z 2 + 2 z 2 a -2 + 3 z 2 a -4 -4 z 2 -3 a 3 z -3 a z + 3 a 4 + 5 a 2 + 3 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (db)

The coefficients of the monomials

t r q j

are shown, along with their alternating sums

χ

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j -2 r = s + 1

j -2 r = s -1

, where

s =

0 is the signature of L9a45. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -1$	$i = 1$
$r = -4$	${\mathbb Z}$	${\mathbb Z}$
$r = -3$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -1$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 0$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2$	${\mathbb Z}^{4}$
$r = 1$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 2$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 3$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$